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Properties

Label 2-690-115.22-c1-0-13
Degree $2$
Conductor $690$
Sign $0.157 + 0.987i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (1.07 − 1.95i)5-s + 1.00·6-s + (−3.43 + 3.43i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−2.14 + 0.624i)10-s − 1.77i·11-s + (−0.707 − 0.707i)12-s + (1.21 − 1.21i)13-s + 4.85·14-s + (0.624 + 2.14i)15-s − 1.00·16-s + (1.12 − 1.12i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.481 − 0.876i)5-s + 0.408·6-s + (−1.29 + 1.29i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.678 + 0.197i)10-s − 0.536i·11-s + (−0.204 − 0.204i)12-s + (0.337 − 0.337i)13-s + 1.29·14-s + (0.161 + 0.554i)15-s − 0.250·16-s + (0.272 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.157 + 0.987i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.157 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620804 - 0.529796i\)
\(L(\frac12)\) \(\approx\) \(0.620804 - 0.529796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.07 + 1.95i)T \)
23 \( 1 + (0.510 + 4.76i)T \)
good7 \( 1 + (3.43 - 3.43i)T - 7iT^{2} \)
11 \( 1 + 1.77iT - 11T^{2} \)
13 \( 1 + (-1.21 + 1.21i)T - 13iT^{2} \)
17 \( 1 + (-1.12 + 1.12i)T - 17iT^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
29 \( 1 + 3.23iT - 29T^{2} \)
31 \( 1 - 6.41T + 31T^{2} \)
37 \( 1 + (0.321 - 0.321i)T - 37iT^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + (6.47 + 6.47i)T + 43iT^{2} \)
47 \( 1 + (-9.10 - 9.10i)T + 47iT^{2} \)
53 \( 1 + (6.42 + 6.42i)T + 53iT^{2} \)
59 \( 1 + 10.4iT - 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + (0.586 - 0.586i)T - 67iT^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + (-3.84 + 3.84i)T - 73iT^{2} \)
79 \( 1 - 6.55T + 79T^{2} \)
83 \( 1 + (-2.56 - 2.56i)T + 83iT^{2} \)
89 \( 1 - 3.44T + 89T^{2} \)
97 \( 1 + (7.79 - 7.79i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01306138749505228547560403343, −9.502777043607578946067081161680, −8.851635680377357255106284304399, −8.060196479209759203877010213119, −6.40723111827490590471992961635, −5.85880613512729733525651447171, −4.84463341182389145446853089718, −3.44620091667159851671102641388, −2.43811356402930807611144209358, −0.60365257619278416800823658508, 1.25340852723115804342521666569, 2.95794561952822636664692568826, 4.16798748824354958968333194873, 5.72402260887299513747533039357, 6.41596671746774414915799526365, 7.17958424117158510162033071461, 7.58283144875400257758871665558, 9.125320480607849175267140589167, 9.972690780026622720205417940263, 10.36341911549747835914769909186

Graph of the $Z$-function along the critical line